bred vector and enso predictability in a hybrid coupled

Bred Vector and ENSO Predictability in a Hybrid Coupled Model during thePeriod 1881–2000

YOUMIN TANG AND ZIWANG DENG

Environmental Science and Engineering, University of Northern British Columbia,

Prince George, British Columbia, Canada

(Manuscript received 27 October 2009, in final form 2 July 2010)

ABSTRACT

In this study, a breeding analysis was conducted for a hybrid coupled El Nino–Southern Oscillation (ENSO)

model that assimilated a historic dataset of sea surface temperature (SST) for the 120 yr between 1881 and

2000. Meanwhile, retrospective ENSO forecasts were performed for the same period. For a given initial state,

15 bred vectors (BVs) of both SST and upper-ocean heat content (HC) were derived. It was found that the

average structure of the 15 BVs was insensitive to the initial states and independent of season and ENSO

phase. The average structure of the BVs shared many features already seen in both the final patterns of

leading singular vectors and the ENSO BVs of other models. However, individual BV patterns were quite

different from case to case. The BV rate (the average cumulative growth rate of BVs) varied seasonally, and

the maximum value appeared at the time when the model ran through the boreal spring and summer. It was

also sensitive to the strength of the ENSO signal (i.e., the stronger ENSO signal, the smaller the BV rate).

Furthermore, ENSO predictability was explored using BV analysis. Emphasis was placed on the re-

lationship between BVs, which are able to characterize potential predictability without requiring observa-

tions, and actual prediction skills, which make use of real observations. The results showed that the relative

entropy, defined using breeding vectors, was a good measure of potential predictability. Large relative en-

tropy often leads to a good prediction skill; however, when the relative entropy was small, the prediction skill

seemed much more variable. At decadal/interdecadal scales, the variations in prediction skills correlated with

relative entropy.

1. Introduction

Intensive research has been focused on studying pre-

dictability, including potential predictability and actual

prediction skill, by examining the perturbation growth.

Currently, there are two methods, the singular vector

(SV) and the bred vector (BV), which are widely used to

analyze the optimal perturbation growth. The earliest

work on SVs can be found in Lorenz’s (1965) paper,

which introduced SVs into meteorology for atmospheric

predictability studies. However, SV analysis was not used

to investigate ENSO predictability until the late 1990s.

Recently, a number of studies on ENSO predictability

have used SVs (e.g., Xue et al. 1997a,b; Chen et al. 1997;

Thompson 1998; Moore and Kleeman 1996, 1997a,b; Fan

et al. 2000; Tang et al. 2006; Zhou et al. 2007).

The breeding method was originally developed to

‘‘bred’’ growing modes related to synoptic baroclinic

instability for short-to-medium-range atmospheric en-

semble forecasting by taking advantage of the fact that

small-amplitude, but faster-growing, convective insta-

bilities saturate earlier. It was first proposed in the 1990s

by Toth and Kalnay (1993, 1997). Breeding vectors are,

by construction, closely related to Lynapunov vectors

(LVs). While BVs have been widely used in mesoscale

and synoptic-scale predictions (e.g., Corazza et al. 2003;

Kalnay 2003; Young and Read 2008), they have rarely

been applied to climate predictions. Cai et al. (2003,

hereafter CKT03) first applied the breeding method

to the Zebiak and Cane (ZC) (1987) ENSO model, and

investigated features of the ENSO BV modes and the

potential of BVs for use in data assimilation and ensem-

ble prediction. Yang et al. (2006, hereafter Y06) explored

the possibility of separating ENSO-related coupled vari-

ability from weather modes and other shorter-time-scale

oceanic instabilities using the breeding technique with

a fully coupled general circulation model (GCM). Some

Corresponding author address: Youmin Tang, Environmental

Science and Engineering, University of Northern British Colum-

bia, 3333 University Way, Prince George, BC V2N 4Z9, Canada.

E-mail: [email protected]

298 J O U R N A L O F C L I M A T E VOLUME 24

DOI: 10.1175/2010JCLI3491.1

� 2011 American Meteorological Society

important properties of ENSO BVs were discovered in

CKT03 and Y06, including the relationship of BVs to the

ENSO background. They also found that a BV pattern is

very similar to the final pattern of the singular vector of

the same coupled model.

Several issues of ENSO BVs still warrant further ex-

ploration. First, both CKT03 and Y06 used model in-

tegration as the reference trajectory (i.e., the initial

state) from which a BV is calculated. They performed

BV analysis continuously during the period of interest

from the beginning to the end. It is also interesting to

explore ENSO BVs by using the actual ENSO evolu-

tions as the reference trajectories, which reveals the

properties and features of ENSO BVs based on realistic

ENSO backgrounds. In particular, the realistic refer-

ence trajectory allows for the exploration of the re-

lationship between BVs, which characterize potential

predictability without requiring observations, and actual

prediction skills, which make use of real observations.

Second, it is well known that ENSO predictability has

a striking decadal/interdecadal variation (e.g., Chen

et al. 2004; Deng and Tang 2009; Tang et al. 2008a). One

might be able to shed some light on the mechanism of

this decadal/interdecadal variation in ENSO predict-

ability by examining temporal variations in potential

predictability, measured by the BV rate or other BV de-

rivatives. In addition, CKT03 and Y06 used, respectively,

an intermediate coupled model and a fully coupled GCM

for ENSO BV analysis. Applying the breeding method

into a hybrid coupled model facilitates ENSO BV analysis

for a full spectrum of models.

Considering the use of the breeding method in a hy-

brid coupled model for the above goals, the period of

BV analysis with a realistic ENSO background should

be long enough to cover sufficient ENSO cycles, allow-

ing statistically robust conclusions to be drawn. In ad-

dition, a long-term retrospective ENSO prediction is

required, in order to examine the relationship between

BVs and actual prediction skill. The relationship between

potential predictability and actual prediction skills has a

practical significance, and may offer a means of estimat-

ing the confidence that we can place in predictions of the

future using the same climate model.

This study pursues the aforementioned objectives. We

recently reconstructed the surface wind stress of the

tropical Pacific back to 1881 using statistical techniques

and using a historic SST and sea level pressure datasets

(Deng and Tang 2009). The reconstructed wind stress

was applied to a hybrid coupled model to perform ret-

rospective ENSO predictions from 1881 to 2000 (Deng

and Tang 2009; Tang et al. 2008a). We also completed

the assimilation of a long-term historic SST dataset into

an oceanic general circulation model (OGCM) that

generated skillful retrospective predictions (Deng et al.

2008). These steps made it possible to implement BVs

with realistic ENSO backgrounds over a long period

and to explore the relationship between BVs and pre-

dictability. This paper is structured as follows: the model

and breeding method are introduced in section 2; the re-

sults on ENSO breeding modes and optimal error growth

rates are presented in section 3; in section 4, the associa-

tion between the features of BVs and ENSO predictability

is explored—especially the measurement of potential

predictability based on BVs; finally, the discussion and

conclusions can be found in section 5.

2. Model and breeding method

a. Model

A hybrid coupled model (HCM) that was composed

of an OGCM coupled to a linear statistical atmospheric

model was used for this study. The OGCM was the latest

version of the Nucleus for European Modeling of the

Ocean (NEMO), identical to the OGCM used in Tang

et al. (2008a) and Deng and Tang (2009). (Details of the

ocean model are available at http://www.lodyc.jussieu.

fr/NEMO/.) The domain of the model used here is

configured for the tropical Pacific Ocean from 308N to

308S and 1228E to 708W, with horizontal resolution 2.08

in the zonal direction and 0.58 within 58 of the equator,

smoothly increasing to 2.08 at 308N and 308S in the me-

ridional direction. There were 31 vertical levels with 17

concentrated in the top 250 m of the ocean. The time step

of integration was 1.5 h and all boundaries were closed,

with no-slip conditions. The statistical atmospheric model

in the HCM was a linear model that predicted the con-

temporaneous surface wind stress anomalies from SST

anomalies (SSTA), which are constructed by the singu-

lar vector decomposition (SVD) method with a cross-

validation scheme. During the initialization of the hybrid

coupled model, the OGCM was forced by the sum of the

associated wind anomalies computed from the atmo-

spheric model and the observed monthly mean climato-

logical wind stress. Full details of the model are given in

Deng et al. (2008) and Tang et al. (2005, 2008a).

To perform long-term hindcasts with the HCM, past

wind stress data are required to initialize forecasts. Us-

ing SST as a predictor, as well as SVD techniques, we

reconstructed a wind stress dataset back to 1881 (Deng

and Tang 2009). The SST data came from the monthly

extended global SST (ERSST) dataset from 1881 to 2000

reconstructed by Smith and Reynolds (2004), and had

28 latitude by 28 longitude resolution. The surface wind

data came from the monthly National Centers for En-

vironmental Prediction–National Center for Atmospheric

1 JANUARY 2011 T A N G A N D D E N G 299

Research (NCEP–NCAR) reanalysis dataset from 1948

to 2000. The data domains for both SST and wind were

configured to the tropical Pacific Ocean. The recon-

structed wind data has been used to study ENSO pre-

dictability (e.g., Tang et al. 2008a; Deng and Tang 2009).

The reconstructed winds were assimilated with SST and

the HCM to produce the initial conditions of predictions

using an ensemble Kalman filter (EnKF; Deng et al. 2008).

The retrospective forecasts for the period from 1881 to

2000 were analyzed in detail in Tang et al. (2008a).

b. Breeding analysis

In this study, we derived a BV using the two-sided self-

breeding algorithm. The nonlinear model is denoted by

M; X indicates the model states that are perturbed for

the BV. The BV at step k 1 1 can be obtained from the

below equations, as schematically described by Fig. 1:

BVk11

5 DX�k11, (1)

DX�k11 51

2

ðtk1t

tk

M(Xk

1 DX1k ) dt

"

�ðt

k1t

tk

M(Xk� DX1

k ) dt

#, (2)

DX1k11 5

DX�k11

lk11

, and (3)

lk11

5B�k11

B1k

, (4)

where k 5 0, 1, . . . , N. The number N is the last rescaling

step. The term lk11 is the rescaling factor at step k 1 1,

which is defined by the amplitude of the perturbation

growth of the upper-ocean 250-m heat content (HC)

over the region of Nino-3.4 (58N–58S, 1708–1208W). The

upper HC was chosen here because of its critical im-

portance in ENSO simulation and prediction (e.g., Tang

2002). However CKT03 found that a BV is, in fact, rel-

atively insensitive to the definition of the rescaling norm.

The variables B�k11 and Bk1 in (4) are the spatially

averaged amplitude of DHC�k11 and DHCk1 of the upper-

ocean 250-m HC over the domain of Nino-3.4, re-

spectively. Note that the HC is only a model diagnostic

variable, thus DHC�k11 is obtained using the HC defini-

tion and (2). In this model, the upper 250 m is covered

by the first 17 layers, and the HC is defined as below

HC 5

�17

i51T

ih

i

�17

i51h

i

, (5)

where Ti and hi are the temperature and thickness, re-

spectively, of the ith model layer obtained by the model

integration [i.e., the first or second item of rhs of (2)] and

the model initial configuration in vertical, respectively.

If we denote DHC by the one-dimensional vector Dhc(i),

i 5 1, 2, . . . , N1, where N1 is the number of model grids

over Nino-3.4 region, we have

B�k11 5

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

N1�N1

i51Dhc2(i)�k11

vuutand (6)

B1k 5

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

N1�N1

i51Dhc2(i)1

k

vuut. (7)

Initially, k 5 0, DX01is the initial random perturbation

x, and B01 is the amplitude of initial perturbation x. The

term t is called the rescaling interval. One round of

process from k to k 1 1 is called one breeding cycle, or

a rescaling step. A steady BV is obtained after a fast

growth of dominant instabilities is saturated (e.g., the

BVs after K 1 L in Fig. 1a).

There are two important parameters in deriving a BV:

the initial perturbation amplitude and t. The initial per-

turbation magnitude for this study was randomly drawn

from a normal distribution with a mean of zero and a

variance of a specified value that was 10% of the variance

of the model heat content anomalies of the upper ocean

250-m HC, that is,

B10 ; N[0, 0.1s(HC)], (8)

where s(HC) is the spatially averaged variance of HC

over the region of Nino-3.4, which was obtained from

the model control run from 1961 to 2000. Thus, the ini-

tial perturbation field x is determined by

x 5 B10 V. (9)

The matrix V is the normalized pseudo-Gaussian

noise field with spatial coherence, given by Evensen

(2003), which has been widely used in ensemble Kalman

filter. Several sensitivity experiments show that, when

using this setting for the initial perturbation, the per-

turbed solution quickly diverges from the control solution

after one rescaling step and meanwhile, the breeding

vectors can converge in four–six rescaling steps, as shown

in Fig. 2.

The rescaling interval is a little arbitrary. Pena and

Kalnay (2004) argued that this period should be longer

than two weeks for slowly varying coupled instability.

CKT03 used three months to obtain bred vectors for the

ZC model whereas Y06 used one month for a coupled


global general circulation model. In this study, we choose

a rescaling period of one month, which leads to the sat-

uration of fast growing instabilities after four–six rescal-

ing steps, as discussed in the next section.

To provide a sufficient sample of the dynamically

important uncertainties of the initial state, we used 15

pairs of random perturbations for each initial condition

(i.e., obtaining 15 BVs). The perturbed variables in-

cluded the sea temperature, horizontal currents, and

vertical currents of the top 250 m (17 model levels).1

The BVs were computed at 3-month intervals from

January 1881 to October 2000 (i.e., January 1881, 1 April

1881, . . . October 2000). Thus there are a total of 15 3

480 BVs for each variable of interest during the entire

period, providing sufficient sampling to conduct robust

statistical analyses. The corresponding time for each BV

was the initial time from which the random initial per-

turbation was added to the model state (see Fig. 1b).

It should be mentioned that the above BV approach is

different from that used in CKT03 and Y06 because they

analyzed BV cycles continuously until the end of the

entire period, as demonstrated in Fig. 1a. The essential

difference between our approach and theirs is the initial

perturbation and initial condition used to generate the

BV. In their approach, the rescaling is done continu-

ously until the end. The initial perturbation for the next

BV was always the previous BV, and was imposed onto

unperturbed forecasts, ensuring the fast growth of in-

stabilities to be saturated in one rescaling step after

a period of initial training (e.g., beyond K 1 L in Fig.

1a). It is similar to the method of Toth and Kalnay

(1993) where the previous BV was centered around the

evolving analysis state for the next BV. A common

feature of CKT03, Y06, and Toth and Kalnay (1993) is

that the analysis state (or unperturbed forecast) and BV

analysis are for the same model. However, the analysis

state in this work is from an ocean data assimilation

system that is different from the coupled model used for

BV analysis; thus, inconsistencies exist between mod-

eling system for the BV cycle and the modeling system

for oceanic analyses, leading to an initial shock in per-

forming BV of each initial condition. Thus, our ap-

proach always takes a complete BV analysis from a

random initial perturbation to the saturation of the fast

FIG. 1. A schematic of the two-sided self-breeding BV algorithm. (a) The steady BVs were

obtained only after a few rescaling steps (i.e., K 1 L) and the unperturbed model forecast was

always used as a reference trajectory for each BV. (b) Each BV was derived from a realistic

initial condition and obtained by six rescaling steps (see text). The interval between two re-

scaling steps (e.g., k to k 1 1) was 1 month. In this study, the BV was analyzed every 3 months

from January 1881 to October 2000 using (b).

1 Considering the computational expense in preparing initial

perturbations, we did not perturb all model variables. The choice of

temperature and currents as the variables that were perturbed was

due primarily to their significant importance in ENSO formulation

and development although it was a somewhat arbitrary. The top

250 m cover the thermocline in the equator, which dominates

ENSO variability.


growth of instabilities for each analysis state (Fig. 1b).

Sensitivity experiments indicate that such a complete

BV analysis usually takes four to six rescaling steps for

the HCM that has a noise-free atmospheric component.

Such an approach also allows us to analyze the BV of

each initial condition in a completely parallel manner. In

concept, our approaches is the same as that of Toth and

Kalnay (1993), including its definition of a BV, although

it is more expensive. We used a realistic analysis state,

rather than an unperturbed forecast, as the initial condi-

tion of each BV because we wanted to explore pertur-

bation growth under a realistic ENSO background, in

addition to the relationship between BV s and prediction

skills. In fact, our approach is very similar to the singular

vector analysis for each initial condition that was per-

formed by Lorenz (1965) using a linear tangent model.

To measure the perturbation growth from the initial

time until the saturation of fast-growing instabilities, a

cumulative growth factor for breeding vector m after step

k was defined, as described in Young and Read (2008):

Gmk 5

lm1 if k 5 1;

lmk Gm

k�1 if k . 1.

�

The term Gkm is lk multiplied by the previous cumu-

lative growth rate. A cumulative growth factor may

better characterize potential predictability because the

prediction errors at leading times of k steps depend on

the error growth not only from k 2 1 to k, but also from

the initial time to k 2 1.

The averaged G6m over 15 BVs is denoted by G6,

namely,

G6

51

15�15

m51Gm

6 . (10)

FIG. 2. Scatterplots of l of two adjacent breeding cycles for all of the perturbations during the entire period from 1881 to 2000. The lines

in (b)–(f) indicate lmk�1 5 lm

k . Note that the x axis is different between (a) and the other panels. The initial perturbation grows very quickly

at the first rescaling step (Fig. 2a) due to the fast growth of nonlinear instabilities; however, the growth rate varies little after four rescaling

steps, indicating a gradual saturation of the growth of nonlinear instabilities.


The term G6 is referred to as the BV rate, the average

growth rate of bred vectors associated with a given ini-

tial state. In total, there are 480 G6 here during the pe-

riod from 1881 to 2000.

3. ENSO BVs and BV rate

In this section, preliminary features of ENSO bred

modes and their variations with initial state, season, and

ENSO signal are explored. The required number of

rescaling steps depends on model dynamics, initial state,

initial perturbation, and other factors. In sensitivity ex-

periments, we found that the fast unstable growth was

saturated after four to six rescaling steps in the HCM.

The scatter-space plot of l of two subsequent rescaling

steps for all of the BVs for the entire period from 1881 to

2000 is shown in Fig. 2. As can be seen in this figure, the

initial perturbation grows very quickly at the first re-

scaling step (Fig. 2a) because of the fast growth of

nonlinear instabilities. However the growth rate varies

little after four rescaling steps, indicating a gradual sat-

uration of the growth of nonlinear instabilities. Thus, we

focused on the BVs of the sixth rescaling step in this

study. Unless otherwise indicated, all of the results of

BVs presented in this paper are from the sixth rescaling

step with the exception of the BV rate that is the cu-

mulative growth rate from rescaling steps 1–6 as defined

above.

a. ENSO bred modes

Figure 3 shows the first EOF modes of BVs for SST

and HC obtained using all of the BVs from 1881 to 2000

(i.e., 480 3 15 BVs), accounting for 61% and 70% of the

total variance, respectively. The first modes, denoted by

EOFtotal, describe a classic ENSO-like structure. Figure

3a for HC has a dipole zonal structure suggesting a

western Pacific ‘‘Rossby wave–like’’ response of one

sign and an eastern Pacific ‘‘Kelvin wave–like’’ response

of the opposite sign (e.g., Tang 2002). Figure 3b for SST

has a large-amplitude signal of one sign located pri-

marily in the equatorial central and eastern Pacific.

These patterns agree with the idea of a heat content

buildup prior to an El Nino as postulated by Jin (1997),

and the delayed oscillator mechanism (Battisti 1988;

Suarez and Schopf 1988); the patterns are also similar

to the BVs from the fully coupled NCEP and the Na-

tional Aeronautics and Space Administration (NASA)

Seasonal-to-Interannual Prediction Project (NSIPP)

GCMs reported in Y06.

When compared with leading singular vectors in many

ENSO models, the patterns in Fig. 3 resemble their final

patterns (e.g., Chen et al. 1997; Xue et al. 1997a; Zhou

et al. 2007; Cheng et al. 2010, hereafter C10). Similarities

between BVs and the final patterns of SVs were also

noticed in CKT03 and Y06.

It has been found in previous work that the SVs are

insensitive to initial conditions in many models (e.g.,

Chen et al. 1997; Xue et al. 1997a; Zhou et al. 2007; C10).

To explore the sensitivity of BV to initial conditions, we

calculated the spatial correlation Rspace between EOFi,

the first EOF mode obtained using 15 BVs for the ith

initial condition (i 5 1, 2, . . . , 480), and EOFtotal, shown

by Fig. 3. If the EOFtotal and the EOFi are denoted by

the normalized one-dimensional vectors eoftotal and eofi,

respectively, the spatial correlation is calculated as fol-

lows:

Rispace 5

1

NP� 1�NP

p51eof

total( p)eof

i( p), (11)

where NP is the number of total model grids over the

model domain, and i 5 1, 2, . . . , 480.

FIG. 3. The first EOF modes of (a) HC and (b) SST. The EOF analysis was employed using all bred vectors from

1881 to 2000. The first modes explain 70% and 61% of the total variance for HC and SST, respectively. The units

are 8C.


The spatial correlation measures the similarity be-

tween the dominant direction in the space spanned by

BVs at different time instances and the dominant di-

rection in the space spanned by BVs computed over

a long time period. The result is shown in Fig. 4. For

most cases (over 90%), the absolute value of the spatial

correlation coefficient is over 0.90, with an overall av-

erage value above 0.85 for all cases for both HC and

SST. As argued by CKT03, the positive phase of the

same pattern should be regarded as equivalent to the

negative phase, at least from a linear system perspective.

In addition, the pattern sign can be arbitrary from the

view of EOF analysis. We also performed bootstrap

experiments [i.e., randomly choose two initial condi-

tions (among 480)], and the spatial correlation of the

leading EOF mode was calculated between the two ini-

tial conditions [i.e., eofj instead of eoftotal in (11)]. The

process was repeated 1000 times. The results showed

that 70% and 93% of the absolute values of the correlation

coefficient were over 0.652 for SST and HC, respectively.

Thus, the preliminary features of BVs, as represented by

the first EOF mode (equivalent to the average), are not

highly sensitive to initial conditions. This conclusion is

further confirmed by Fig. 5, the pattern of the first EOF

modes obtained using all of the BVs in four different

calendar months, indicating the seasonal independence of

the BVs.

The above finding on the insensitivity of BVs to ini-

tial conditions results from the similarity of the EOF-

characterized dominant structure in the space spanned

by the BVs of different initial conditions. Further exami-

nation was conducted on individual BVs at different initial

conditions. In other words, is an individual BV at the

FIG. 4. The spatial correlation between EOFtotal and EOFi, where EOFtotal is the first EOF mode obtained using all

BV vectors (480 3 15), whereas EOFi is the first EOF mode obtained using 15 BVs for the initial condition i (i 5 1,

2, . . . , 480).

2 The value of 0.65 was chosen arbitrarily, a very large value

compared with the threshold value of statistical significance at the

confidence level of 99%.


initial time t1 similar to another BV at its initial time t2? To

explore this question, we randomly chose two individual

BVs at two different initial conditions and calculated their

spatial correlation coefficient. This process was repeated

1000 times, as shown in Fig. 6. The correlation coefficients

are almost evenly distributed across the range from the

maximum (1) to the minimum (21), indicating that an

individual BV is dependent on its initial conditions.

The average of the ensemble BVs had a stable struc-

ture independent of the initial state whereas the in-

dividual member of ensemble BVs was sensitive to the

initial state. One physical interpretation for this is that

the mean of ensemble BVs is probably dominated by

time-independent model dynamics (e.g., the delayed-

oscillator mechanism), whereas the individual BV mem-

ber is greatly impacted by an initial random perturbation.

In Toth and Kalnay’s approach, the initial perturbation

came from multiple modes orthogonalizing in various

ways that were not completely random in nature; thus, the

BV derived from their approach is probably equivalent to

the mean of the ensemble of BVs here.

b. Variations of BV rate

The BV rate G6, defined in section 2b, measures the

mean growth rate of all of the BVs for a given initial

state. Shown in Fig. 7 is the average G6 of all of the BVs

for four different calendar months. As can be seen, the

greatest BV rates occur in January and April, during

which the model runs through the boreal spring and

summer (because of the 6 rescaling steps for each BV,

equivalent to 6 months). This is in good agreement with

the variation of the optimal error growth rate in the ZC

model (e.g., CKT03; C10). The seasonal dependence of

error growth might explain why ENSO prediction skill

often drops remarkably when predictions go through the

spring (i.e., the ‘‘spring barrier’’). This increase in error

is due to the fact that the intertropical convergence zone

(ITCZ) is closest to the equator during the spring, sus-

taining unstable conditions and enhancing convective

instability. In contrast, the ocean–atmosphere interac-

tion is strong during the summer due to the relatively

large vertical temperature gradient and ocean upwelling

(e.g., Xue et al. 1997a), which lead to strong baroclinic

instability.

Because of the existence of decadal/interdecadal

variations in ENSO variability and prediction skills (e.g.,

Tang et al. 2008a), it is interesting to examine temporal

variations in the BV rate. Figures 8a,b show variations in

the BV rate and the Nino-3.4 SSTA index from 1881 to

2000, for which a 2-yr running mean has been applied to

FIG. 5. The first EOF mode of HC obtained using all of the BVs of four different calendar months.


address interannual and longer signals. As previously

indicated, the interannual variability is obvious in both

the BV rate and in the Nino-3.4 SSTA index. Further

scrutiny of Figs. 8a,b reveals that the interannual vari-

ability of the BV rate has decadal/interdecadal varia-

tion. For example, the magnitude of the BV rate is

visibly smaller during the 1880s and 1980s than during

other periods. The variability of the BV rate at various

time scales is more obvious in the wavelet analysis

shown in Fig. 8c. The local wavelet power spectrum

clearly indicates that significant periods were localized

in time and varied from 2 to 16 yr from 1881 to 2000. A

striking feature in Figs. 8c,d is the strong interannual

variability and relatively weak decadal variability. Com-

paring Figs. 8c with 8d reveals similar oscillatory char-

acteristics between the BV rate and ENSO variability,

suggesting an association of the BV rate with the ENSO

background.

The dependence of the growth rate of BVs on the

ENSO phase has been addressed in CKT03 and Y06.

They found that a large error growth tends to occur in

a neutral or onset/breakdown stage of an ENSO event,

such as 3–4 months prior to the peak phase, whereas

a small error growth often occurs in an ENSO peak

phase. In their work, the model integration was used as

the reference trajectory for their BV analysis, as men-

tioned in section 2b. The relationship between the BV

rate and ENSO phase was then explored by comparing

our BVs with the realistic ENSO evolution as the initial

reference trajectory.

Figure 9 shows the average BV rate as a function of

ENSO phase. The ENSO events are binned into nine

categories based on the Nino-3.4 SSTA index. Three

kinds of events, El Nino, La Nina, and neutral events,

are explored here. They are classified by Nino-3.4 SSTA

index (8C) greater than 2.0, 1.5, and 1.0; less than 22.0,

21.5, and 1.0; and in the range of (20.5, 0.5), (20.3, 0.3),

and (20.1, 0.1), respectively. Considering 6 rescaling

steps (equivalent to 6 months) for each BV, we selected

the BV rate of the 3 months prior to the time when the

FIG. 6. Correlation between two arbitrary BVs from two arbitrary initial conditions. These arbitrary BVs and initial

conditions were randomly chosen, and this process was repeated 1000 times.


criteria value was met. Thus, the BV rate shown in

Fig. 9 actually represents the error growth of a 6-month

duration (i.e., 3-month prior to and after the occur-

rence of an ENSO event). As can be seen in Fig. 9,

there was a small error growth rate during the peak

ENSO stages (El Nino and La Nina), while a large

error growth rate occurred during the neutral stages.

The stronger the background SSTA of ENSO, the

smaller the BV rate. These results are generally con-

sistent with those in CKT03 and Y06 as well as former

SV studies (e.g., Chen et al. 1997; Zhou et al. 2007).

Both CKT03 and Zhou et al. 2007 analytically derived

this converse relationship between the error growth

rate and the ENSO signal by using a delay-oscillator

model (Suarez and Schopf 1988). This relationship

suggests that a strong ENSO signal tends to reside in

a dynamical stable regime, whereas a weak signal is

probably more associated with an instable dynamical

process. Another possibility is that the model ENSO

strength is bounded above so that error growth can take

place only in one direction (e.g., neutral phase) when

one is near the extreme, resulting in a situation where

instability might remain large in the direction toward

neutral. In addition, Fig. 9 shows that the BV rate is

smaller during El Nino than during La Nina, possibly

because the tropical wave instability is more active

when the cold tongue is well established in the east

(Y06).

The converse relationship between the BV rate and

ENSO signal is further demonstrated in Fig. 10, the

scatterplot of the BV rate against the ENSO signal. The

ENSO signal here is defined by the variance of the ob-

served Nino-3.4 SSTA index, computed using a 20-yr

running window from 1881 to 2000.3 A strong converse

relationship is observed in Fig. 10 with a correlation

coefficient of 20.71 between the ENSO signal and the

BV rate. A similar result can be obtained using other

measures of the ENSO signal, such as the total spectrum

power of Nino-3.4 SSTA index at the frequencies of

2–5 yr or the variance explained by the first EOF mode

of SSTA (HCA).

FIG. 7. The average BV rate for all of the initial conditions for

January, April, July, and October.

FIG. 8. Variation in (a) the BV rate and (b) the Nino-3.4 SSTA

index from 1881 to 2000. The wavelet power spectrum of (c) the BV

rate and (d) the Nino-3.4 SSTA using the Morlet wavelet. The thick

contour encloses regions of greater than 95% confidence using

a red-noise background spectrum. The solid smooth curves in the

left and right corners indicate that edge effects become important.

The time window used here was 2ffiffiffi2p

T, a function of period T.

3 The variance obtained during a 20-yr window is plotted at the

middle point of the window. For example, the variance at 1890 is

calculated using the samples from 1881 to 1900. The 20-yr window

is shifted by 3-months each time starting from 1881 to 2000. The BV

rate plotted is the average value of BVs of each window.


4. ENSO predictability

In this section, we will explore ENSO predictability.

Emphasis was placed on identifying a measure calcu-

lated from BVs that was able to quantify potential pre-

dictability and had a relationship with actual prediction

skill. This relationship offers a practical means of esti-

mating the confidence level of an ENSO forecast using

the HCM.

The retrospective ENSO prediction was performed for

the period from 1881 to 2000 using the HCM. A total of

480 forecasts, initialized from January 1881 to October

2000, were run starting at 3-month intervals (1 January,

1 April, 1 July, and 1 October), and continued for 12 months

for the HCM. The SST assimilation was performed to

initialize the forecasts, as introduced in section 2a.

a. Bred vector and BV rate

Figure 11 shows the average correlation R and RMSE

of SSTA predictions against the observed values over

lead times of 1–6 months [i.e., (1/6)�6

l51Rl(RMSE

l),

where l is the index of lead month] for the period from

1881 to 2000. As indicated, the best correlations occur in

the equatorial central and eastern Pacific (as in many

ENSO models; e.g., Goddard et al. 2001; Tang et al.

2008a), where there was the largest error growth, as

shown in Fig. 3. This seemingly contradicts the wide-

spread perception that good prediction ability should

correspond with a small error growth rate (e.g., Farrell

1990; Tang et al. 2008b; Orrell 2003). One probable

reason is that error growth, as shown in Fig. 3, can be

decomposed into linear growth and nonlinear growth.

The linear errors only impact amplitude prediction,

whereas nonlinear errors impact phase prediction. Thus,

the correlation is still likely to be high if the large error

growth rate is due mainly to linear instabilities. Tang and

Deng (2010) found that, indeed, the region of high cor-

relation coefficients has relatively weak nonlinearity. This

argument is also supported by the fact that the region of

good correlations often has a large RMSE in ENSO

models, as shown in Fig. 11b.

Figure 12 shows the scatterplot of the correlation and

the RMSE of individual predictions against the corre-

sponding BV rates using the same initial conditions ap-

plied to both the BVs and the predictions. The RMSE/

correlation value of individual predictions (RMSEIP/

CIP) was used to measure the error (correlation) of an

individual forecast for all leads L (512 months), defined as

RMSEIP 5

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

L� 1�t5L

t51[Tp(t)� To(t)]2

vuutand (12)

CIP 5

�L

t51(T

pt � mp)(To

t � mo)ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�L

t51(T

pt � mp)2

vuutffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�L

i51(To

t � mo)2

vuut. (13)

Here, T is the index of Nino-3.4 SSTA, p denotes

forecasts, o denotes observations, t is the lead time of

the forecast, mp is the mean of the forecasts, mo is the

FIG. 9. The average BV rate as a function of the ENSO phase,

indicating the error growth rate during La Nina, neutral, and

El Nino events. Three kinds of El Nino (La Nina) cases were ex-

plored with Nino-3.4 SSTA index greater (less) than 1.0 (21.0), 1.5

(21.5), and 2.0 (22.0), respectively; whereas three kinds of neutral

cases were explored using the ranges of Nino-3.4 SSTA index be-

tween (20.5, 0.5), (20.3, 0.3), and (20.1, 0.1), respectively.

FIG. 10. Scatterplot of the BV rate against the ENSO signal.


mean of observations, and L is the number of samples

used.

Figure 12 includes all of the initial conditions from

1881 to 2000, showing little visible relationship between

the BV rate and prediction skill and suggesting that the

BV rate probably is not a good measure of potential

predictability for this HCM. However, one should take

caution when interpreting this figure because of the

small sample size used in making this calculation, es-

pecially for the correlation coefficient. We next explore

the other measure of potential predictability and ex-

amine its relationship to prediction skills.

b. Relative entropy derived from BVs

Recently, a new theoretical framework for measuring

the uncertainty of predictions has been developed and

applied to examine atmospheric and climate potential

predictabilities (Schneider and Griffies 1999; Kleeman

2002, 2008; Tippett et al. 2004; Tang et al. 2005, 2008b;

DelSole 2004, 2005). The approach is built on informa-

tion theory (Cover and Thomas 1991). It has been argued

that the relative entropy (RE) of an individual prediction,

defined as the difference between the climatological

probability density function (PDF) and the prediction

PDF, is a good measure of its prediction utility. A larger

RE indicates that more useful information is being sup-

plied by the prediction, which could be interpreted as

making it more reliable.

In the case where the PDFs are Gaussian, which is

approximately the case in many practical instances (in-

cluding ENSO prediction), the relative entropy RE of

an individual prediction may be expressed exactly in

terms of the prediction variance sp2, evaluated typically

by ensemble members, the model climatological vari-

ance sq2, obtained from a model climatological run, and

the difference mp 2 mq of the ensemble and the clima-

tological means (Kleeman 2002):

RE 51

2log

s2q

s2p

1s2

p

s2q

1(mp � mq)2

s2q

� 1

" #. (14)

Figure 13 is as in Fig. 12, but using the RE of Nino-3.4

SSTA index instead of the BV rate. Here the climato-

logical variance and mean were obtained from samples

of all BVs for all initial conditions from 1881 to 2000

(i.e., 480 3 15 samples), as in Tippett et al. (2004) and

Tang et al. (2005). As can be seen in Fig. 13, a large RE

often led to a good prediction skill (i.e., a small RMSE

and a large correlation); however, when the RE was

small, the prediction skill seemed much more variable.

This result is very similar to the so-called triangular re-

lationship that has been found to characterize the re-

lationship between ensemble spread and prediction

skill in ensemble weather forecast. This finding also re-

sembled the relationship between ensemble signal and

prediction skill in ensemble climate prediction (ECP;

e.g., Buizza and Palmer 1998; Xue et al. 1997b; Moore

and Kleeman 1998; Tang et al. 2008b). For instance, it

was found in ECP that, when the ensemble signal is

large, the prediction is invariably good; whereas when it

is small, the prediction can be much more variable. Thus,

we also used the ‘‘triangular relationship’’ to describe

the relationship between the RE and prediction accu-

racy in this study. Figure 13 suggests that the RE derived

from BVs could be an effective indicator of ENSO pre-

diction skill.

The importance of RE in measuring potential pre-

dictability has been addressed in previous studies (e.g.,

Kleeman 2002, 2008; Tippett et al. 2004; Tang et al. 2005,

2008b), but all of these used prediction ensemble to

FIG. 11. (a) Correlations and (b) RMSEs of predicted Nino-3.4

SSTA against the observed values for the past 120 yr from 1881 to

2000, averaged over lead times of 1–6 months.


calculate RE. Here we used BVs to calculate RE and

obtained similar results. This is probably due to the in-

trinsic coherence of BVs with ensemble predictions.

From the point of calculation expense, the BV-based

RE is comparable to the ensemble-prediction-based RE

when the BV analysis is incorporated with the ensemble

run. However, if the ensemble prediction system is in-

dependent to the generation of the optimal perturbation

modes (e.g., singular vector), the BV-based RE is much

cheaper than the ensemble-prediction-based RE.

c. Decadal/interdecadal variation in relative entropyand predictability

In many ENSO models, ENSO predictability clearly

has a decadal/interdecadal variation (e.g., Chen et al.

2004; Tang et al. 2008a). It was found that the inter-

decadal variation in ENSO predictability was in good

agreement with the signal of interannual variability

and with the degree of asymmetry of an ENSO system

(e.g., Tang et al. 2008a; Tang and Deng 2010). Below

the relationship between relative entropy and ENSO

predictability at decadal/interdecadal time scales is

examined.

Figure 14 shows the average R and RMSE values for

the Nino-3.4 SSTA index with a 1–12-month lead,

measured by a 20-yr running window that moved at 1-yr

intervals from 1881 to 2000 (i.e., 1881–1900, 1882–1901,

1883–1902, . . . , 1980–99, 1981–2000), where the predic-

ted Nino-3.4 SSTA is compared against the observed

value.4 The average relative entropy over the running

window is also plotted in Fig. 14 (solid line). As can be

seen in this figure, the prediction skills show a pro-

nounced interdecadal variation. Generally there was a

high prediction skill in the late nineteenth century and in

the middle to late twentieth century, and a low skill from

1901 to 1960. Such an interdecadal variation feature was

also manifested in the RE. The correlations between the

RE and the correlation coefficient and between the RE

and the RMSE were 0.76 and 20.66, respectively, thus,

the larger the RE, the larger the correlation and the

smaller the RMSE.

Further examination of RE might provide some in-

sights into interdecadal variations in ENSO predict-

ability. The first two terms on the right-hand side (rhs) of

(14) are determined by the climatological variance and

predictive variance. Because variance of climatology is

time invariant, these terms represent a measurement of

FIG. 12. Scatterplot of (a) correlations and (b) RMSEs of each individual prediction against the BV rate for the

corresponding initial condition.

4 The average skills, as a function of the index of window, were

obtained by (1/12)�12

l 5 1RLl (RMSEL

l ), where l is the index of the

lead month and L is an index of a 20-yr window that shifts by 1 yr

from the beginning to the end during the period from 1881 to 2000

(January 1881, April 1881, . . . , October 2000). The term RlL(RMSEl

L)

is the skill for the window L measured using 20-yr samples, as a

function of the lead time l.


the dispersion or spread of the BVs. The third term on

the rhs of (14) is governed by the amplitude of the mean

of the BVs. We refer to the first two terms minus 1 as the

dispersion component (DC) and the third term as the

signal component (SC).

Figure 15 shows the variations in DC and SC against

RE for the period from 1881 to 2000. This figure reveals

that DC is significantly larger than SC, and RE is dom-

inated by DC. As can be seen in Fig. 15a, RE and DC

vary linearly with a slope of unity. The correlation co-

efficient between R and DC is 0.97. In contrast to the

good relationship between DC and RE, the relation

between SC and RE was much less significant. Figure 13

indicates that the triangular relationship between the

RE and prediction skill is equivalent to the triangular

relationship between ensemble spread and skill that was

found in some numerical weather models.

It should be noticed that Fig. 15 differs from an earlier

result reported by Tang et al. (2005). They found that

the SC plays a dominant role in the RE s for two ENSO

models. This is most likely due to disparities with models

and methods used in these two cases. In this study, we

used a different HCM. In particular, BVs were used to

calculate the RE rather than the ensemble prediction

itself used in Tang et al. (2005). Figures 15 and 14 in-

dicate a possible mechanism of interdecadal variation in

ENSO predictability; namely, when the ENSO cycle

resides in a phase that has a relatively large DC, it is

more predictable and vice versa.

5. Discussion and summary

The breeding method has been widely applied in a

variety of fields including atmospheric and climate pre-

dictability, data assimilation, and ensemble prediction.

FIG. 13. Scatterplot of (a) correlations and (b) RMSEs of each individual prediction against the relative entropy for

the corresponding initial condition.

FIG. 14. Variations in the relative entropy (solid), the average

correlation coefficient (dashed), and RMSE (dotted) over 1–

12-month lead times from 1881 to 1990. Normalization was done

for each variable prior to plotting.


In this study, we applied the breeding method to a hy-

brid coupled model and demonstrated the feasibility of

using BVs to facilitate ENSO predictability studies.

Specifically we performed, for the first time, each BV

starting from a realistic ENSO background for the past

120 yr from 1881 to 2000. Meanwhile, retrospective

forecasts of the past 120 yr were also conducted. The

absence of stochastic atmospheric transients in this hy-

brid coupled model facilitated the ability of the breeding

method to identify slowly growing error modes. The

results show that the breeding method was able to detect

climatically relevant coupled instabilities present in this

coupled model.

The characteristics of ENSO BVs and BV rates were

analyzed. It was found that the preliminary BV mode

had an ENSO-like structure with a dipole zonal struc-

ture between the western Pacific and the eastern Pacific

in the subsurface heat content field and a large-amplitude

signal in the equatorial central and eastern Pacific. This

structure is similar to the final pattern responding to the

first singular vector and also reminiscent of the physical

processes controlled by the delayed-oscillator mecha-

nism. The dominant structure is insensitive to initial

conditions like singular vectors in many ENSO models.

However, the spatial structure of each individual BV is

sensitive to the initial conditions.

The seasonal dependence of the error growth rate

(BV rate) was investigated. The largest BV rates oc-

curred when the model runs through the boreal spring

and summer (i.e., spring barrier). The BV rate was also

strongly dependent on the ENSO phase. A small error

growth rate occurred during the peak ENSO stages,

whereas a large error growth rate occurred during the

neutral stages. A significant converse relationship exis-

ted between the signal of the background SSTA and the

error growth rate: the stronger the background SSTA of

ENSO, the smaller the BV rate, and vice versa. This

converse relationship is primarily due to the nonlinear

relationship (e.g., exponential) between the signal and

growth rate, as indicated by theoretical analyses (CKT03;

Zhou et al. 2007).

To explore ENSO predictability, we conducted ret-

rospective ENSO predictions for the period from 1881

to 2000. The BV rate was analyzed in terms of its ability

to measure the model prediction skill. It was found that

an overall relationship between the BV rate and pre-

diction skill was absent in this model. Furthermore, we

calculated and analyzed the relative entropy from BVs.

The relative entropy was found to be an effective in-

dicator of prediction utility. The relationship between

the relative entropy and prediction skill varies at a vari-

ety of time scales. For individual cases, it appeared to be

like a ‘‘triangular relationship’’: large relative entropy

often led to a good prediction skill; however, when the

relative entropy was small, the prediction skill seemed

much more variable. At decadal/interdecadal scales, there

was consistent variation in prediction skill and in relative

entropy. In the late-nineteenth and the late-twentieth

FIG. 15. The relationship of the (a) dispersion component and the (b) signal component to RE.


centuries, the relative entropy was large and the model

had a high correlation and low RMSE. During other

periods, the relative entropy was small, and the model

showed the poor prediction skill. Furthermore, it was

found that relative entropy was dominated by the spread

of BVs, whereas the mean of BVs had little contribution

to RE.

The coupled model used in this study was a hybrid

coupled in absence of the fastest-growth weather in-

stabilities inherent to stochastic atmospheric transients.

While such a hybrid model facilitates the detection of

ENSO error modes by using the breeding method, it

lacks some necessary physical and dynamical processes

that exist in the real world. It would be interesting to

validate the results presented in this study with com-

plicated GCM models, a line of research that we will

pursue in the future. In addition, the window length of

20 yr used in this study is slightly arbitrary and was

motivated by previous work (e.g., Chen et al. 2004; Tang

et al. 2008a). Nevertheless, this study analyzed ENSO

BVs using realistic ENSO backgrounds as reference

trajectories, and, in particular, explored the relationship

between BVs and actual prediction skill for a long pe-

riod of over 100 yr. The results offer valuable insight to

ENSO predictability and have practical significance to

ensemble prediction.

Acknowledgments. This work is supported by the Ca-

nadian Foundation for Climate and Atmospheric Sci-

ences (CFCAS) through Grant GR-7027 and the Canada

Research Chair program. We thank the anonymous re-

viewers for their constructive comments, which helped to

improve this manuscript.

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